2. Relevant equations
Well, I know that we can change this using a double angle rule, so that the integrals become 1/2 + 1/2 cos (2*pi/2n)(x1 + x2 +... xn))dx1 dx2 ... dxn

and the integral over the 1/2 just becomes 1/2, but the other side baffles me.

3. The attempt at a solution
My professor tried to do this, but I don't agree with his methodology. When he integrated it, he got pi/n out front, and if you keep integrating, this would go to pi^n / n^n . However, the integral should produce n^n / pi^n if I'm not mistaken, meaning this would diverge to infinity, and not go to zero like he said. Any ideas?

and the integral over the 1/2 just becomes 1/2, but the other side baffles me.

3. The attempt at a solution
My professor tried to do this, but I don't agree with his methodology. When he integrated it, he got pi/n out front, and if you keep integrating, this would go to pi^n / n^n . However, the integral should produce n^n / pi^n if I'm not mistaken, meaning this would diverge to infinity, and not go to zero like he said. Any ideas?

2. Relevant equations
Well, I know that we can change this using a double angle rule, so that the integrals become 1/2 + 1/2 cos (2*pi/2n)(x1 + x2 +... xn))dx1 dx2 ... dxn

and the integral over the 1/2 just becomes 1/2, but the other side baffles me.

3. The attempt at a solution
My professor tried to do this, but I don't agree with his methodology. When he integrated it, he got pi/n out front, and if you keep integrating, this would go to pi^n / n^n . However, the integral should produce n^n / pi^n if I'm not mistaken, meaning this would diverge to infinity, and not go to zero like he said. Any ideas?

It might help to use the fact that ##\cos(w)## is the real part of ##e^{iw}##, so you need to integrate
[tex] \int_0^1 \int_0^1 \cdots \int_0^1 e^{i c (x_1 + x_2 + \cdots + x_n)} \, dx_1 \, dx_2 \cdots \, dx_n,[/tex]
where ##c = \pi/n##. The exponential factors into separate factors for each variable, and so the n-fold integration is just the product of single-variable integrals. You can pull the "real part" operator outside all the integrals (why?).